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Loose Engel structures

Published online by Cambridge University Press:  06 January 2020

Roger Casals
University of California Davis, Department of Mathematics, Shields Avenue, Davis, CA 95616, USA email
Álvaro del Pino
Utrecht University, Department of Mathematics, Budapestlaan 6, 3584 CD Utrecht, The Netherlands email
Francisco Presas
Instituto de Ciencias Matemáticas – CSIC, C. Nicolás Cabrera 13–15, 28049 Madrid, Spain email


This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$-principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

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© The Authors 2020

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Borman, M. S., Eliashberg, Y. and Murphy, E., Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), 281361.CrossRefGoogle Scholar
Bryant, R. L. and Hsu, L., Rigidity of integral curves of rank 2 distributions, Invent. Math. 114 (1993), 435461.CrossRefGoogle Scholar
Cartan, E., Sur quelques quadratures dont l’élément différentiel contient des fonctions arbitraires, Bull. Soc. Math. France 29 (1901), 118130.10.24033/bsmf.639CrossRefGoogle Scholar
Cartan, É., Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205241.CrossRefGoogle Scholar
Casals, R., Murphy, E. and Presas, F., Geometric criteria for overtwistedness, J. Amer. Math. Soc. 32 (2019), 563604.CrossRefGoogle Scholar
Casals, R. and del Pino, Á., Classification of Engel knots, Math. Ann. 371 (2018), 391404.10.1007/s00208-017-1625-0CrossRefGoogle Scholar
Casals, R., Pérez, J., del Pino, Á. and Presas, F., Existence h-principle for Engel structures, Invent. Math. 210 (2017), 417451.10.1007/s00222-017-0732-6CrossRefGoogle Scholar
Colin, V., Presas, F. and Vogel, T., Notes on open book decompositions for Engel structures, Algebr. Geom. Topol. 18 (2018), 42754303.CrossRefGoogle Scholar
del Pino, Á., On the classification of prolongations up to Engel homotopy, Proc. Amer. Math. Soc. 146 (2018), 891907.CrossRefGoogle Scholar
del Pino, Á. and Vogel, T., The Engel-Lutz twist and overtwisted Engel structures, Preprint (2018), arXiv:1712.09286.Google Scholar
Eliashberg, Y., Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623637.CrossRefGoogle Scholar
Eliashberg, Y., Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier 42 (1992), 165192.CrossRefGoogle Scholar
Eliashberg, Y. and Mishachev, N., Introduction to the h-principle, Graduate Studies in Mathematics, vol. 48 (American Mathematical Society, Providence, RI, 2002).Google Scholar
Geiges, H., An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
Gershkovich, V. Ya. and Vershik, A. M., Nonholonomic dynamical systems. Geometry of distributions and variational problems, in Dynamical systems – 7, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., vol. 16 (VINITI, Moscow, 1987), 585.Google Scholar
Gromov, M., Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 9 (Springer, Berlin, 1986).CrossRefGoogle Scholar
Kotschick, D. and Vogel, T., Engel structures and weakly hyperbolic flows on four-manifolds, Comment. Math. Helv. 93 (2018), 475491.CrossRefGoogle Scholar
Little, J. A., Nondegenerate homotopies of curves on the unit 2-sphere, J. Diff. Geom. 4 (1970), 339348.CrossRefGoogle Scholar
Mitsumatsu, Y., Geometry and dynamics of Engel structures, Preprint (2018),arXiv:1804.09471.Google Scholar
Moerdijk, I. and Mrčun, J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91 (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Montgomery, R., Generic distributions and Lie algebras of vector fields, J. Differential Equations 103 (1993), 387393.CrossRefGoogle Scholar
Montgomery, R., Engel deformations and contact structures, in Northern California symplectic geometry seminar, Advances in the Mathematical Sciences, vol. 45 (American Mathematical Society, Providence, RI, 1999), 103117; American Mathematical Society Transl. Ser. 2, 196.Google Scholar
Montgomery, R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91 (American Mathematical Society, Providence, RI, 2002).Google Scholar
Montgomery, R. and Zhitomirskii, M., Points and curves in the Monster tower, Memoirs of the American Mathematical Society, vol. 956 (American Mathematical Society, Providence, RI, 2009).Google Scholar
Peralta-Salas, D., del Pino, Á. and Presas, F., Foliated vector fields without periodic orbits, Israel J. Math. 214 (2016), 443462.CrossRefGoogle Scholar
Pia, N., Riemannian properties of Engel structures, Preprint (2019), arXiv:1905.09006.Google Scholar
Saldanha, N., The homotopy type of spaces of locally convex curves in the sphere, Geom. Topol. 19 (2015), 11551203.CrossRefGoogle Scholar
Thurston, W., The theory of foliations of codimension greater than one, Comm. Math. Helv. (1974), 214231.CrossRefGoogle Scholar
Thurston, W., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), 249268.CrossRefGoogle Scholar
Vogel, T., Existence of Engel structures, Ann. of Math. (2) 169 (2009), 79137.CrossRefGoogle Scholar
Vogel, T., Non-loose unknots, overtwisted discs and the contact mapping class group of 𝕊3, Geom. Funct. Anal. 28 (2018), 228288.CrossRefGoogle Scholar
Yamazaki, K., Engel manifolds and contact 3-orbifolds, Preprint (2018), arXiv:1811.09076.Google Scholar
Zhao, Z., Lagrangian Engel structures, Preprint (2018), arXiv:1805.09147.Google Scholar
Zhao, Z., Complex and Lagrangian Engel structures. PhD thesis, Duke University (2018).Google Scholar
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